All of the 3rd grade teachers and students from Oak went on a field trip to an art museum. Tickets were $$7.00$ each for teachers and $$2.50$ each for students, and the group paid $$43.00$ in total. The next month, the same group visited a science museum where the tickets cost $$14.00$ each for teachers and $$7.50$ each for students, and the group paid $$101.00$ in total. Find the number of teachers and students on the field trips.
Solution: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${7x+2.5y = 43}$ ${14x+7.5y = 101}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-2$ ${-14x-5y = -86}$ ${14x+7.5y = 101}$ Add the top and bottom equations together. $ 2.5y = 15 $ $ y = \dfrac{15}{2.5}$ ${y = 6}$ Now that you know ${y = 6}$ , plug it back into $ {7x+2.5y = 43}$ to find $x$ ${7x + 2.5}{(6)}{= 43}$ $7x+15 = 43$ $7x = 28$ $x = \dfrac{28}{7}$ ${x = 4}$ You can also plug ${y = 6}$ into $ {14x+7.5y = 101}$ and get the same answer for $x$ ${14x + 7.5}{(6)}{= 101}$ ${x = 4}$ There were $4$ teachers and $6$ students on the field trips.